Package 'elfDistr'

Kumaraswamy Complementary Weibull Geometric (Kw-CWG) Probability Distribution

Description

Density, distribution function, quantile function and random generation for the Kumaraswamy Complementary Weibull Geometric probability distribution (Kw-CWG) lifetime distribution.

Details

This package follows naming convention that is consistent with base R, where density (or probability mass) functions, distribution functions, quantile functions and random generation functions names are followed by d, p, q, and r prefixes.

Behaviour of the functions is consistent with base R, where for not valid parameters values NaN's are returned, while for values beyond function support 0's are returned (e.g. for non-integers in discrete distributions, or for negative values in functions with non-negative support).

All the functions vectorized and coded in C++ using Rcpp.

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Kumaraswamy Complementary Weibull Geometric Probability Distribution

Description

Density, distribution function, quantile function and random generation for the Kumaraswamy Complementary Weibull Geometric (Kw-CWG) probability distribution.

Usage

dkwcwg(x, alpha, beta, gamma, a, b, log = FALSE)

pkwcwg(q, alpha, beta, gamma, a, b, lower.tail = TRUE, log.p = FALSE)

qkwcwg(p, alpha, beta, gamma, a, b, lower.tail = TRUE, log.p = FALSE)

rkwcwg(n, alpha, beta, gamma, a, b)

Arguments

x, q	vector of quantiles.
alpha, beta, gamma, a, b	Parameters of the distribution. 0 < alpha < 1, and the other parameters mustb e positive.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

$f(x) = \alpha^a \beta \gamma a b (\gamma x)^{\beta - 1} \exp[-(\gamma x)^\beta] \cdot \frac{\{1 - \exp[-(\gamma x)^\beta]\}^{a-1}}{\{ \alpha + (1 - \alpha) \exp[-(\gamma x)^\beta] \}^{a+1}} \cdot$

$\cdot \bigg\{ 1 - \frac{\alpha^a[1 - \exp[-(\gamma x)^\beta]]^a}{\{ \alpha + (1 - \alpha) \exp[-(\gamma x)^\beta] \}^a} \bigg\}$

Cumulative density function

$F(x) = 1 - \bigg\{ 1 - \bigg[ \frac{\alpha (1 - \exp[-(\gamma x)^\beta]) }{ \alpha + (1 - \alpha) \exp[-(\gamma x)^\beta] } \bigg]^a \bigg\}^b$

Quantile function

$Q(u) = \gamma^{-1} \bigg\{ \log\bigg[\frac{ \alpha + (1 - \alpha) \sqrt[a]{1 - \sqrt[b]{1 - u} } }{ \alpha (1 - \sqrt[a]{1 - \sqrt[b]{1 - u} } ) }\bigg] \bigg\}^{1/\beta}, 0 < u < 1$

References

Afify, A.Z., Cordeiro, G.M., Butt, N.S., Ortega, E.M. and Suzuki, A.K. (2017). A new lifetime model with variable shapes for the hazard rate. Brazilian Journal of Probability and Statistics

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